# Constitutive Model

The poly crystal plasticity model formulated by Asaro and Needleman (1985) is employed in the analyses. Accordingly, the total deformation of a crystallite is taken to be the result of two distinct physical mechanisms: crystallographic slip due to dislocation motion on the active slip systems, and elastic lattice distortion. Within an FCC crystal, plastic deformation occurs by crystallographic slip on the 12{1П}<110> slip systems. For a BCC crystal, crystallographic slip is assumed to occur on 24 slip systems, the 12{110}<111> + 12{112}<111> systems.

In the rate-sensitive crystal plasticity model employed, the elastic constitutive equation for each crystal is specified by:

v.

a = LD — o’0 – otrD (1)

V

where О is the Jaumann rate of Cauchy stress, D represents the strain-rate tensor and L is the tensor

of elastic moduli. The term O0 is a viscoplastic type stress-rate that is determined by the slip rates on the slip systems of a FCC and BCC crystal. A detailed presentation of the crystal plasticity constitutive model can be found in Wu et al. (1997) and will not be repeated here.

The slip rates are governed by the power-law expression

where ^(0) is a reference shear rate taken to be the same for all the slip systems, т^а-) is the resolved shear stress on slip system a, g(a) is its hardness and m is the strain-rate sensitivity index. The g(a)

characterize the current strain-hardened state of all the slip systems. The rate of increase of the function g(a) is defined by the hardening law:

where g(a) (0) is the initial hardness, taken to be a constant r0 for each slip system, and where the h(a/3) values are the hardening moduli. The form of the moduli is given by

h(afi) = Ч(аЮЪ(Ю (no sum on P’)

where h(p) is a single slip hardening rate and q^ap) is the matrix describing the latent hardening behaviour of the crystallite.

The single slip hardening law employed in this investigation takes the following power-law form of the function h( p)

where h0 is the system’s initial hardening rate, n is the hardening exponent and у a is the accumulated slip.

Two different models are employed to obtain the response of a polycrystal comprised of many grains. In the Taylor model, the material response is obtained by invoking the Taylor assumption. Accordingly, at a material point representing a polycrystal of N grains, the deformation in each grain is taken to be identical to the macroscopic deformation of the continuum. Furthermore, the macroscopic values of all quantities, such as stresses, stress-rates and elastic moduli, are obtained by averaging their respective values over the total number of grains at the particular material point. In the FE/grain model an element of the finite element mesh represents a single crystal, and the constitutive response at a material point is given by the single crystal constitutive model. This approach enforces equilibrium and compatibility between grains throughout the polycrystalline aggregate in the weak finite element sense.